Proxy ensemble geometric phase and proxy index of time-reversal invariant topological insulators at finite temperatures

TL;DR

This paper introduces proxy topological indices derived from the transfer matrix to effectively characterize finite-temperature topological phases in time-reversal invariant insulators, providing a computationally efficient alternative to existing methods.

Contribution

It proposes proxy index and proxy ensemble geometric phase directly from the transfer matrix, extending topological characterization to finite temperatures without splitting contributions.

Findings

Proxy index and proxy EGP smoothly extend ground-state topological indices to finite temperatures.

Comparison with Uhlmann phase reveals different temperature transition behaviors.

Proxy EGP does not require splitting contributions, simplifying calculations.

Abstract

The ensemble geometric phase (EGP) has been proposed as a topological indicator for finite-temperatures systems. The ensemble Wilson loop, or the transfer matrix, contains the crucial information in the EGP construction. We propose a proxy index and a proxy EGP directly from the transfer matrix and apply them to time-reversal invariant topological insulators exemplified by the Bernevig-Hughes-Zhang (BHZ) and Kane-Mele (KM) models. The quantized proxy index and proxy EGP smoothly generalize the ground-state topological index to finite temperatures. For the BHZ model, a comparison with another topological indicator, the Uhlmann phase, shows different transition behavior with temperature. For the KM model, the EGP have been generalized to the time-reversal EGP previously, but the proxy EGP does not require any splitting of the contributions. The proxy index and proxy EGP thus offer an efficient means for characterizing finite-temperature topological properties.